What is the solution set for the inequation lx+1/x l>2?

Q. What is the solution set for the inequation |x+1/x |>2?

First of all from the question you’ve written it is not clear if |x+1/x | means

Anyway we’ll discuss both of them.


1.  
Note that inside the modulus we are simply adding a number to its inverse. Sum of any positive number and its multiplicative inverse is always greater than 2 unless that number is 2. This can be proved by AM  GM. Also if the inequality is true for any positive k then it is also true for –k as well.
Hence solution set will be R – {–1. 1}.

 

2.

Let . So the inequality becomes

For t > –1:

For t < –1:
–1 – t > 2

Hence,

 



 

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First of all from the question you’ve written it is not clear if |x+1/x | means


Anyway we’ll discuss both of them.


1.  
Note that inside the modulus we are simply adding a number to its inverse. 
Sum of any positive number and its multiplicative inverse is always greater than 2 unless that number is 2. 
This can be proved by AM  GM. 
Also if the inequality is true for any positive k then it is also true for –k as well.
Hence solution set will be R – {–1, 1} or all real numbers except 1 and -1.

 

2.

Let . So the inequality becomes

For t > –1:

For t < –1:
–1 – t > 2

Hence,

 

 

 



 

  • 1
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